Answer to Question #297222 in Microeconomics for Heena

Question

Given the two factor product in Q= 150L^0.5 K^ 0.5, wage rate of labour = 50 and rental cost of capital = 40. Determine the amount of labour and capital that will minimize the cost of producing 1118 units of output?

Answer 1

$Q=150L^{0.5}K^{0.5}$

$C=50L+40K$

Fitting in the langragian equation:

$l=50L+40K-h(1118-150L^{0.5}K^{0.5})$

Differentiate the langragian equation with respect to L and K:

$\frac{dl}{dL}=50-h75^{-0.5}K^{0.5}=0......(i)$

$\frac{dl}{dK}=50-h75L^{0.5}K^{-0.5}=0.....(ii)$

Equating equation (i) and (ii):

$50-h75L^{-0.5}K^{0.5}=50-h75^{0.5}K^{-0.5}$

Like terms on both sides cancel each other:

$L^{-0.5}K^{0.5}=L^{0.5}K^{-0.5}$

$\frac{K^{0.5}}{L^{0.5}}=\frac{L^{0.5}}{K^{0.5}}$

Therefore $K=L$

Replacing K and Q in

$Q=150L^{0.5}K^{0.5}$

We get:

$1119=150L^{0.5}L^{0.5}$

$1118=150L^{2}$

$L=2.7$

Replacing L we get:

$1118=150×(2.7)^{0.5}K^{0.5}$

$4.54=√K$

$K=2.13$